Rotational Motion Slide 2 / 37 Angular Quantities An angle can be - - PowerPoint PPT Presentation

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Rotational Motion

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Angular Quantities

An angle θ can be given by: where r is the radius and l is the arc length. This gives θ in radians. There are 360° in a circle or 2π radians. 360° = 2π rad 1 rad = 57.3°

r r

l

θ

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1 A 0.25 m arc in a circle of radius 5 m is being studied. How many radians is this?

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2 A 0.25 m arc in a circle of radius 5 m is being studied. How many degrees is this?

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Angular Quantities

Just like velocity is the change in displacement over the change in time, angular velocity is defined as the change in angle over the change in time. Likewise, angular acceleration is the change in angular velocity over the change in time.

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3 An object moves π/2 radians in 8 seconds. What is it's angular velocity?

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4 An object starts from rest and accelerates to an angular velocity of 16 rad/s in 10 seconds. What is it's angular acceleration?

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Angular Quantities

Linear velocity can be related to angular velocity. The linear velocity of something moving in a circle would be the change in arc length over the change in time. Using θ = l/r, the change in length would be Δl = rΔθ. And we already defined angular velocity as Δθ/Δt.

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5 What is the linear speed of a child on a merry-go- round of radius 3 m that has an angular velocity of 4 rad/s?

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6 What is the angular velocity of an object traveling in a circle of radius 0.75 m with a linear speed of 3.5 m/s?

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Angular Quantities

Similarly, tangential acceleration can also be related to angular acceleration. But there is also radial acceleration, which we know to be v2/r but this can also be related to angular velocity.

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The linear acceleration is the vector sum of the angular and radial accelerations. alinear = atan + aR

Angular Quantities

atan aR alinear

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7 A child pushes a merry-go-round with radius of 2.5 m from rest to an angular velocity of 3 rad/s in 8

• seconds. What is the angular acceleration?

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8 What is the merry-go-round's tangential acceleration at that time?

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9 What is the merry-go-round's radial acceleration at that time?

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10 What is the merry-go-round's linear acceleration at that time?

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We know from circular motion that frequency is defined as the number of revolutions per second. So frequency can be given in terms of angular velocity. And angular velocity can be given in terms of frequency.

Angular Quantities

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11 What is the frequency of the merry-go-round in the previous problem?

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Just like the for kinematics equations for linear motion...

Kinematics Equations & Rolling Motion

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The same four exist for rolling motion...

Kinematics Equations & Rolling Motion

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12 A bicycle wheel with a radius of 0.3 m starts from rest and accelerates at a rate of 4.5 rad/s2 for 11 seconds. What is it's final angular velocity?

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13 A bicycle wheel with a radius of 0.3 m starts from rest and accelerates at a rate of 4.5 rad/s2 for 11 seconds. What is it's final linear velocity?

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14 A bicycle wheel with a radius of 0.3 m starts from rest and accelerates at a rate of 4.5 rad/s2 for 11 seconds. What is it's angular displacement during that time?

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15 A bicycle wheel with a radius of 0.3 m starts from rest and accelerates at a rate of 4.5 rad/s2 for 11 seconds. How many revolutions did it make during that time?

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16 A bicycle wheel with a radius of 0.3 m starts from rest and accelerates at a rate of 4.5 rad/s2 for 11 seconds. What is it's linear displacement during that time?

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17 A 50 cm diameter wheel accelerates from 5 revolutions per second to 7 revolutions per second in 8 seconds. What is the angular acceleration?

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18 A 50 cm diameter wheel accelerates from 5 revolutions per second to 7 revolutions per second in 8 seconds. What is the angular displacement during that time?

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19 A 50 cm diameter wheel accelerates from 5 revolutions per second to 7 revolutions per second in 8 seconds. How far will a point on the wheel have traveled during that time?

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Up to this point we described rotational motion without addressing the cause of rotational motion. The dynamics of rotational motion is analogous to the dynamics of linear motion, where force is replaced by torque. Torque is defined as force x lever arm (perpendicular).

Torque

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20 A force of 300 N is applied to a crowbar 20 cm from the axis of rotation. Calculate the torque.

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21 Now if that same force is applied to the same crow bar but now at a 25o angle. Calculate the torque.

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22 A student wants to balances two weights of 2 kg and 0.5 kg on a meter stick. How far does she have to put the fulcrum from the 2 kg weight?

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In linear motion, a force causes an acceleration. In rotational motion, a torque causes an angular acceleration. F = ma F = mrα since a = rα τ = mr2α since τ = Fr we can multiply both sides by r This is just for a single particle. For an entire body we take the sum of all the torques: Στ = Σ(mr2)α Where mr2 is called the moment of inertia, I. Στ = Iα

Dynamics

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Here are some moments of inertia for common objects.

Moments of Inertia

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Remember that kinetic energy is ½mv2. This is for an object that is undergoing translational motion. For an object undergoing rotational motion Kinetic Energy can be given as: KE = ½mv2 KE = ½m(ωr)2 since v = ωr KE = ½(mr2)ω2 rotational KE = ½Iω2

Rotational Kinetic Energy

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Since W = Fdperpendicular... W = FΔl W = FrΔθ since θ = l/r W = τΔθ since τ = Fr

Work

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Just like we have conservation of momentum and energy, we also can use the principle of conservation of momentum. Angular momentum is analogous to linear momentum. p = ma L = Iω The total angular momentum of an object is constant if the net torque on the object is zero.

Conservation of Angular Momentum